Precalculus Examples

$5^{- \frac{t}{2}} = 0.2$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (5^{- \frac{t}{2}}) = \ln (0.2)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (5^{- \frac{t}{2}})$ by moving $- \frac{t}{2}$ outside the logarithm.

$- \frac{t}{2} \ln (5) = \ln (0.2)$

Step 2.2

Combine $\ln (5)$ and $\frac{t}{2}$ .

$- \frac{\ln (5) t}{2} = \ln (0.2)$

Step 3

Simplify the left side.

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Step 3.1

Reorder factors in $- \frac{\ln (5) t}{2}$ .

$- \frac{t \ln (5)}{2} = \ln (0.2)$

Step 4

Multiply both sides of the equation by $- \frac{2}{\ln (5)}$ .

$- \frac{2}{\ln (5)} (- \frac{t \ln (5)}{2}) = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \frac{2}{\ln (5)} (- \frac{t \ln (5)}{2})$ .

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Step 5.1.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{2}{\ln (5)}$ into the numerator.

$\frac{- 2}{\ln (5)} (- \frac{t \ln (5)}{2}) = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.1.2

Move the leading negative in $- \frac{t \ln (5)}{2}$ into the numerator.

$\frac{- 2}{\ln (5)} \cdot \frac{- t \ln (5)}{2} = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{\ln (5)} \cdot \frac{- t \ln (5)}{2} = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{\ln (5)} \cdot \frac{- t \ln (5)}{2} = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{\ln (5)} (- t \ln (5)) = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.2

Cancel the common factor of $\ln (5)$ .

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Step 5.1.1.2.1

Factor $\ln (5)$ out of $- t \ln (5)$ .

$\frac{- 1}{\ln (5)} (\ln (5) (- t)) = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{\ln (5)} (\ln (5) (- t)) = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.2.3

Rewrite the expression.

$- - t = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 t = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.1.1.3.2

Multiply $t$ by $1$ .

$t = - \frac{2}{\ln (5)} \ln (0.2)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{2}{\ln (5)} \ln (0.2)$ .

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Step 5.2.1.1

Combine $\ln (0.2)$ and $\frac{2}{\ln (5)}$ .

$t = - \frac{\ln (0.2) \cdot 2}{\ln (5)}$

Step 5.2.1.2

Move $2$ to the left of $\ln (0.2)$ .

$t = - \frac{2 \ln (0.2)}{\ln (5)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{2 \ln (0.2)}{\ln (5)}$

Decimal Form:

$t = 2$